Higher Engineering Mathematics, Sixth Edition

578 Higher Engineering Mathematics

x

A

C

D

B

y

300

250

200

150

100

100 200 300

Frequency in hertz

Inductive reactance in ohms

400 500

50

0

Figure 60.2

lines of best fit. A graph showing co-ordinate values is

called ascatter diagramin statistics.

Problem 4. The experimental values relating

centripetal force and radius, for a mass travelling at

constant velocity in a circle, are as shown:

Force (N) 5 10 15 20 25 30 35 40

Radius (cm) 55 30 16 12 11 9 7 5

Determine the equations of (a) the regression line of

force on radius and (b) the regression line of radius

on force. Hence, calculate the force at a radius of

40cm and the radius corresponding to a force of

32newtons.

Let the radius be the independent variableX,andthe

force be the dependent variableY. (This decision is

usually based on a ‘cause’ corresponding toXand an

‘effect’ corresponding toY.)

(a) The equation of the regression line of force on

radius is of the formY=a 0 +a 1 Xand the con-

stantsa 0 anda 1 are determined from the normal

equations:

∑

Y=a 0 N+a 1

∑

X

and

∑

XY =a 0

∑

X+a 1

∑

X^2

(from equations (1) and (2))

Using a tabular approach to determine the values

of the summations gives:

Radius,X Force,Y X^2

55 5 3025

30 10 900

16 15 256

12 20 144

11 25 121

9 30 81

7 35 49

5 40 25

∑

X= 145

∑

Y= 180

∑

X^2 = 4601

XY Y^2

275 25

300 100

240 225

240 400

275 625

270 900

245 1225

200 1600

∑

XY= 2045

∑

Y^2 = 5100

Thus 180= 8 a 0 + 145 a 1

and 2045= 145 a 0 + 4601 a 1

Solving these simultaneous equations gives

a 0 = 33 .7anda 1 =− 0 .617, correct to 3 signifi-

cant figures. Thus the equation of the regression

line of force on radius is:

Y=33.7−0.617X

(b) The equation of the regression line of radius on

force is of the formX=b 0 +b 1 Yand the con-

stantsb 0 andb 1 are determined from the normal

equations: